From ae61ae229fd5007c91bffc7038f83f51fc9ab7d6 Mon Sep 17 00:00:00 2001
From: allen <allen@5c0f84ea-6048-4d6e-bfa4-55cd5f2e0dd7>
Date: Tue, 14 May 2002 16:47:25 +0000
Subject: Added documentation for plane waves, still needs to be checked
 carefully with the source code and parameters

git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDLinearWaves/trunk@73 5c0f84ea-6048-4d6e-bfa4-55cd5f2e0dd7
---
 doc/documentation.tex | 63 ++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 45 insertions(+), 18 deletions(-)

diff --git a/doc/documentation.tex b/doc/documentation.tex
index 8386fc3..c8d5081 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -12,32 +12,59 @@ Mark Miller, Malcolm Tobias, Paul Walker}
 
 \section{Purpose}
 
-There are two different linearized initial data sets provided:
+There are two different linearized initial data sets provided, plane waves 
+and Teukolsky waves.
 
-\begin{enumerate}
-\item	plane waves \\
-Plane waves cane be specified to be travelling in an arbitrary direction.
-The form of the wave packet is: 
+\section{Plane Waves}
+
+A full description of plane waves can be found in the PhD Thesis of 
+Malcolm Tobias, {\it The Numerical Evolution of Gravitational Waves}, 
+which can be found at {\tt http://wugrav.wustl.edu/Papers/Thesis97/Thesis97.html}.
+
+Plane waves travelling in arbitrary directions can be specified.  For
+these plane waves the TT gauge is assumed (the metric perturbations
+are transverse to the direction of propagation, and the metric is
+traceless). In the case of waves travelling along the $z-$direction
+this would give the {\it plus} solution
+$$
+h_{xx}=-h_{yy}=f(t\pm z), h_{xy}=h_{xz}=h_{yz}=h_{zz} = 0
+$$
+and the {\it cross} solution
+$$
+h_{xy}=h_{yx}=f(t\pm z), h_{yz}=h_{xx}=h_{yy}=h_{zz}=0
+$$
+This thorn implements the {\tt plus} solution, with the waveform
+$f(t\pm z)$ having the form of a Gaussian modulated sine function. 
+Now working with a general direction of propagation $k$ we have the
+plane wave solution:
+$$
+f(t,x,y,z) = A_{in} e^{-(k_i^p x^i + \omega_p(t-r_a) )^2} \cos(k_ix^i+\omega t)
+     + A_{out} e^{-(k_i^p x^i -\omega_p(t-r_a))^2} \cos(k_i x^i - \omega t)
+$$
+and
+\begin{eqnarray*}
+g_{xx}&=& 1 + f[\cos^2\phi - \cos^\theta\sin^2\phi]
+\\
+g_{xy}&=& - f \sin^2 \theta \sin \phi \cos \phi
+\\
+g_{xz} &=& f \sin\theta \cos\theta \sin\phi
+\\
+g_{yy} &=& 1+f [\sin^2\phi - cos^2\theta \cos^2\phi]
+\\
+g_{yz} &=& f \sin\theta \cos\theta \cos\phi
+\\
+g_{zz} &=& 1-f\sin^2\theta
+\end{eqnarray*}
+The extrinsic curvature is then calculated from 
 \begin{equation}
- A*exp\left[-(kp_ix^i-\omega_p (time-ra))^2\right]
- cos(k_ix^i-\omega \ time),
+K_{ij} = - \frac{1}{2\alpha} \dot{g}_{ij}
 \end{equation}
-where:\\
-A = amplitude of the wave  \\
-k  = the wave number of the sine wave \\
-$\omega$ = the frequency of the sine wave  \\
-kp = the wave number of the gaussian modulating the sine wave \\
-$\omega_p$ = the frequency of the gaussian \\
-ra = the initial position of the packet(s). \\
 
-
-\item	Teukolsky waves \\
+\section{Teukolsky waves}
 Teukolsky waves are quadrupole wave solutions to the linearized
 Einstein equations.  For a full description, see: PRD 26:745 (1982).
 
 
-\end{enumerate}
-
 \section{Comments}
 The extrinsic curvature is initialized assuming the initial lapse is one.
 
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